Optimal Transport Domain Adaptation (OTDA)

Optimal Transport provides a geometric framework for aligning probability distributions by minimizing the cost of moving mass from one distribution to another [^1]. When applied to domain adaptation, OT learns a transport map that transforms data from a source domain (e.g., one scanner site) to match the distribution of a target domain (e.g., a reference site).

Unlike ComBat-based methods that assume parametric location-scale shifts, OTDA makes no distributional assumptions and can align arbitrarily complex site differences, including non-linear distortions and multi-modal distributions.

Method

OTDA finds a coupling matrix \(\mathbf{P}\) that minimizes transport cost between source and target samples:

\[\min_{\mathbf{P}} \langle \mathbf{P}, \mathbf{C} \rangle + \lambda \cdot \text{regularization}(\mathbf{P})\]

where \(\mathbf{C}\) is the cost matrix (e.g., Euclidean distance) and \(\lambda\) controls regularization (e.g., entropic smoothing for Sinkhorn).

After fitting \(\mathbf{P}\), source samples are transformed via barycentric projection:

\[\mathbf{X}_{\text{transformed}} = \text{diag}(\mathbf{P} \mathbf{1})^{-1} \mathbf{P} \mathbf{X}_{\text{target}}\]

Key features

Aspect	Detail
Flexible transport	Supports EMD, Sinkhorn, Sinkhorn with group Lasso, and Laplace regularization
Supervised mode	Uses labels to guide same-class transport (lower cost between matched classes)
Multi-reference	Can align to multiple reference sites combined
Distribution-free	No normality or linearity assumptions

Parameters

Parameter	Options	Default	Description
ot_method	“emd”, “sinkhorn”/”s”, “sinkhorn_gl”/”s_gl”, “emd_laplace”/”emd_l”	“emd”	Transport algorithm
metric	Any scipy distance metric	“euclidean”	Cost function
reg	float	1.0	Entropic regularization (Sinkhorn)
eta	float	0.1	Group Lasso regularization
limit_max	int or None	10	Sets infinite cost for different classes (semi-supervised)

Example

from uniharmony.ot import OptimalTransportDomainAdaptation

otda = OptimalTransportDomainAdaptation(
    ot_method="sinkhorn",
    metric="sqeuclidean",
    reg=0.1
)

# Fit with reference site
otda.fit(X_train, sites_train, ref_site="site_A", y=labels_train)

# Harmonize new data
X_harmonized = otda.transform(X_test, sites_test)